Algebra III
Lesson 8
Statements of Similarity
– Proportional Segments –
Angle Bisectors and Side Ratios
Statement of Similarity
B
E
C
A
D
F
∆ ABC ~ ∆FED
Letters must follow in exact order
∆ CBA ~ ∆ DEF
∆ ABC ~ ∆ DFE
(Not similar)
Setting up ratios from statement of similarity
∆ ABC ~ ∆ FED
AB
FE
BC
ED
AC
FD
Example 8.1
In ∆ ABC, segment BD is the altitude to the hypotenuse AC
AB
AD
Show the following:
=
AC
AB
B
A
C
D
Make a statement of similarity
∆ ADB ~ ∆ ABC
Get the ratios
AD
AB
AB
AC
DB
BC
AD
=
AB
AB
AC
Proportional segments ( reproving something already doing)
Two triangles that are similar
a
c
b
x
y
a+b
c
a
c+d
d
x
Get ratios of:
a+b
a
c+d
c
y
y
x
c+d
a+b
=
c
a
c d
a b
+
= +
c c
a a
1+
d
b
= 1+
c
a
b
a
=
d
c
Example 8.2
Find x
x+1
x+4
13
19
x +1
=
13
x+4
19
19 ( x + 1) = 13 ( x + 4 )
19 x + 19 = 13 x + 52
6 x = 33
x =
33
6
=
11
2
Angle Bisectors and side ratios
Start with a triangle
Bisect one angle
a
b
c
d
The two stars emphasize that the angle has been bisected.
So you can make a ratio of sides in two ways
a b
=
c d
or
a
c
=
b d
Example 8.3
In ∆ ABC, segment BD is the angle bisector of angle B
Find x
B
10
A
15
x
D
12
x
12
=
10 15
15 x = 120
x=8
C
Example 8.4
a)
In ∆ ABC, segment AD is the angle bisector of angle A
Find x
B
6
D
x
A
14
20
x
6
=
20 14
14 x = 120
x=
=
120
14
60
7
C
Practice
a) In ∆ ABC, BD is the angle bisector of angle B
B
Find x
8
A
12
x
8
D
x
8
=
8 12
12 x = 64
x=
=
64
12
16
3
C
b) Given ∆ XYZ is a right triangle YM is the altitude to the hypotenuse of XZ
Y
Show the following:
XY
YZ
=
XM
MY
Make two similar triangles
X
∆ XMY ~ ∆ YMZ
Set up ratios
XM
XY
MY
YZ
XY
XZ
Flip them over
XY
XM
YZ
MY
XZ
XY
Now pick two needed
XY
YZ
=
XM
MY
M
z
c) Is the following argument valid or invalid?
All authors wear gold watches
John wears a gold watch
John is an author
Invalid:
.
Follows major premise backwards